@heather another thing: "And the development of those laws is still not really complete" what do you mean by this exactly? "Quantum mechanics" is not really a fundamental law of its own, but more or a general recipe for how to make theories. I wouldn't say its development is "incomplete". Sure, better theories could come in the future to replace it, but that does not mean that the current theory is not complete

Say you cook up a model about a physical system. Such a model consists of, say, a system of differential equations. What criterion decides whether the model is classical or quantum-mechanical?
None of the following criteria are valid:
Partial differential equations: Both the Maxwell equations ...

There are two famous cases in classical mechanics that fail to be deterministic.
The first, and most famous, is Norton's Dome, which corresponds to a system with a force of the form
$$F = \sqrt{r} $$
There are more details on the Wikipedia article (it's usually described as the result of a rea...

0celo and Slereah were having an extensive discussion regarding that issue a couple of months back

Locality is another very subtle issue and I wouldn't really take it for granted

@Blue Aside from having to use relativistic physics to treat this properly, infinite force implies the system has an infinite energy after any negligibly small amount of time (or has infinite mass, which is a black hole, which does require GR+QM)

@Mithrandir24601 Okay. So we're putting some observable physical constraints on it (apparently): 1) Point particles don't exist 2) The universe is relativistic and some others. While that's very reasonable, I find it philosophically unsatisfactory that we should restrict ourselves to the laws of physics which are observable for our universe and neglect the other mathematically legitimate possibilities. Anyhow, thanks for the explanation!

At some point I'd like to learn the exact mathematical conditions for these things though, they're probably quite subtle

I don't know if there exist legitimate models of non-relavistic or non-quantum mechanical universes though

@Blue I'm ignoring point 2 for simplicity. If you want something classical with infinite mass, then that's much worse in that the gravitational force on everything in the universe will be infinite, so everything will either have infinite energy or will be contained in some infinitely massive object somewhere

At the beginning you have a system Ax=b. If a is Hamiltonian (its self adjoint) then you know that exp(iAt) is unitary and so can be simulated by a quantum circuit. Hamiltonian simulation is just the application of a quantum circuit that implements/approximates the effect of the unitary matrix exp(iAt) on the quantum state

@Nelimee I sort of get that part. However, I do not understand how we'd create a control on several qubits (which are in the control register). Do we have to break up $e^{iAt}$ into controlled-by-single-qubit gates?

And we use Hamiltonian simulation because the exp(iAt) matrix has eigenvalues that are related to A's eigenvalues (if L_i are the eigenvalues of A, exp(itL_i) are the eigenvalues of exp(iAt)), and from the shape of the eigenvalues of exp(iAt) we can see that the phase estimation algorithm will allow us to recover the eigenvalue of A from the simulation of exp(iAt) (Hamiltonian simulation)

@Nelimee And the action of all those individual single-qubit controlled unitary gates is equivalent to $\sum_{\tau = 0}^{T-1}|\tau\rangle \langle \tau|^{C}\otimes e^{iA\tau t_0/T}$? I not very sure how to show that they're equivalent

And I know that it works like that (for the sum), but I don't feel confident enough with it to explain it to someone else :/ I'll read a little more in a book I have, and see if I can have a clear explanation on that part

But the $\left|\tau\right>\left<\tau\right|$ is the "control" part, i.e. this means that the operation will be controlled by the register $C$.

After the QPE, the state will be: 1. Exactly the state you gave $\left( \sum_{j=1}^N \beta_j \left|u_j\right> \otimes \left|\tilde\lambda_j\right>\right)$ if the QPE had enough qubits to encode the eigenvalues in the register. 2. A state "close" to $\left( \sum_{j=1}^N \beta_j \left|u_j\right> \otimes \left|\tilde\lambda_j\right>\right)$ if the QPE hadn't enough qubits to encode the eigenvalues in the register.

You can look at the QISKit implementation but if you have a gate $G$ composed of $n$ simpler gates $g_i$, then the inverse of $G$ ($G^\dagger$) is obtained by taking each gate composing $G$ in reverse order and inverse them one by one

Because that was my idea at the beginning, but Hamiltonian simulation for a general matrix is not even done in theoretical papers, and Hamiltonian simulation for specific matrices (sparse for example) is REALLY challenging

@Nelimee not directly. I'm currently more on the side of classical ML applied to QM now, but I am trying to get a deeper understanding of QC and in particular QML algorithms. I would like to also go in that direction in the future. I started looking at those papers when I started my PhD but didn't get much out of it. I find kind of appalling how cryptic that whole area is

though it's probably just because no one understands it well enough to clear the waters

anyway, fully understanding HHL is kind of a dream of mine at this point lol

I want to know what time complexity is considered efficient/inefficient for quantum computers. For this, I need to know how many operations a quantum computer can perform per second. ჩan anyone tell me how to calculate it and what factors it depends on (implementation details or number of qubits ...

@Blue it's probably not that hard to go through the details, no. Although getting through and understanding the details still doesn't fully qualify as "fully understanding" for me. I feel confident in saying I "fully understand" something only when it feels "totally obvious" and intuitive. Indeed, I have yet to understand anything in QC

The governments, big companies (list of quantum processors) and smaller ones are in the competition of building bigger and bigger quantum computers.
Not unexpectedly the number of qubits of those quantum computers seem to double every year but those qubits are noisy qubits. What is a more meani...

I don't understand very well the second question in quantumcomputing.stackexchange.com/q/2393/55. Isn't the procedure the other way around? They "magically" have $|\Psi_0\rangle|b\rangle$, and they apply that conditional operation to make it into the state that you say is after the phase estimation step

I can't help you on $R(\tilde \lambda^{-1})$ part, I did not understand it correctly myself :/ I just implemented it as they present it in the 4x4 paper, and it wasn't working until I magically changed a swap before the rotations. So for the moment, I don't even know why my implementation is working ><

For a Hilbert space $\mathcal{H}_A$, I have seen the phrase
density matrices acting of $\mathcal{H}_A$
multiple times, e.g. here.
It is clear to me that if $\mathcal{H}_A$ is finite ($|A|=n$), then this makes sense mathematically, because a density matrix $\rho$ can be written as $\rho \i...

Theorem 2 of [1] states:
Suppose $C$ is an additive self-orthogonal sub-code of $\textrm{GF}(4)^n$, containing $2^{n-k}$ vectors, such that there are no vectors of weight $<d$ in $C^\perp/C$. Then any eigenspace of $\phi^{-1}(C)$ is an additive quantum-error-correcting code with parameters $[...

@AndrewO not very much to be honest. We're thinking about giving the feeds to this room better names but that's about it. Also, there were issues with @Blue's chat account recently meaning the event got deleted and recreated, so you might want to resubscribe

would you agree @Mithrandir24601 that the key difference between classical and quantum probabilities (or one of them) is the way via superposition quantum probabilities affect each other? along with measurement

@Blue waves interfere. I don't think you're justified in saying that 'quantum interference' is what makes quantum special as you're just moving the argument to 'what's special about quantum interference?'

Like many quantum computation researchers, back when I was first learning the basics of the field, I relied heavily on Nielsen & Chuang's "Quantum computation and quantum information" textbook.
However, one frustrating aspect of doing so was that no official set of solutions was ever released fo...

@Blue at its heart, it's still a variety of wave interference. Double slit is wave interference. If you show me an experiment and go 'look, interference!', I'm not going to assume that you've got a quantum system

@Blue if what you mean by quantum interference is e.g. the thing that causes antibunching, then yeah, that's non-classical. It also violates an inequality describing the boundary between classical and quantum physics :P

@Mithrandir24601 I never said that's not good intuition but my point was it introduces misconceptions in people like "operators" are "infinite dimensional matrices" and "infinite dimensional vectors" are analogous to "infinite dimensional arrows". While my nitpicking is indeed irritating, I say this precisely because I struggled a lot due to these exact misconceptions (which various introductory level books seem to contain)

Heather and everybody else is surely free to write at whatever depth they wish to. However, I'm also free to mention my views. And so are you free to oppose them. ;)

@heather Honestly though, do you really understand what you wrote about Bell's inequality violation, as in, it makes sense to you? Because it doesn't make sense to me. It's one of the more subtle topics and I'd really like a discussion about that when you're free. It's one of those topics which I'm not confident about

This is a good question and in my view gets at the heart of a qubit. Like the comment by @blue its not that it can be a equal superposition as this is the same as a classical probability distribution. It is that it can have negative signs.
Take this example. Imagine you have a bit in the $0$ st...

Upon reading carefully, this one makes more sense to me

Although I still have a few confusions looming

@heather Is it possible to relate your Bell inequality argument with the transition matrix argument (for any number of qubits) ?

@Semiclassical Ah, you're here at the right time. Any views on the relation between Bell's inequalities and quantum interference (as in the transition matrix written in that answer)?

@Blue There comes a point when you look at something and start going "hmm... I'm not sure about this... Is that right? I want more detail here". This is how things are - there's a fundamental point at which science goes "it could be this, this, this or even this but we're just not sure" and you kind of have to live with that, in much the same way that every year you'll have to live with getting told "what you knew last year was wrong" - this *never* stops. Many answers and published papers have this 'problem'. It's just finding out where the 'problem' is. Like Siereah's link currently on th…

Although @heather have you considered making headings in your answer to make it easier to read?